The point is to handle duplications, if all letters are unique the answer is simply (26 over 8)

  DB<288> sub fac {my $x=1; $x*=$_ for 2..$_[0]; $x}

  DB<289> sub binom { my ($n,$k)=@_; fac($n)/(fac($n-$k)*fac($k)) }

  DB<290> binom 26,8
 => 1562275
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Reasoning is simple, it calculates all binary vectors of length 26 with exactly 8 1-bits.

But with duplications its more complicated, e.g. 4 out of "aabcd" is not (5 over 4)=5

a bcd
 abcd
aa cd
aab d
aabc
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cause the first 2 solutions are identical.

Generating all combinations and filtering the unique once is normally not very clever, cause the overhead can be enormous.

  DB<292> length "aaaabbbcccddddeeeeffffgggghhhiiijjkkkllllmmmnnnnoooo
+pppqrrrrssssstttuuuuvvwwwxxyyyzzzz"
 => 86

  DB<293> binom 86,8
 => "53060358690"
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And I'm stuck finding a formula which calculates all unique solutions, but generating is easier, just don't allow bitvectors with "gaps" between identical letters:

so

 "aaaabbbc..." # pattern
 "1000100...." # ok      => ab...
 "1100100...." # ok      => aab...
 "1001100....  # not ok  => aab...
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I will update this post with a loop generating all possibilities soon.

update

indeed L~R's number of 12461993 possibilities is correct

The following (non-optimized) code took 3 minutes to calculate them:

use strict;
use warnings;
use Data::Dump qw/pp/;



my %max=(
     a => 4,
     b => 3,
     c => 3,
     d => 4,
     e => 4,
     f => 4,
     g => 4,
     h => 3,
     i => 3,
     j => 2,
     k => 3,
     l => 4,
     m => 3,
     n => 4,
     o => 4,
     p => 3,
     q => 1,
     r => 4,
     s => 5,
     t => 3,
     u => 4,
     v => 2,
     w => 3,
     x => 2,
     y => 3,
     z => 4,
    );
my $maxword=8;

#%max=(a=>2,b=>1,c=>1,d=>1);
#$maxword=4;

my @keysort=sort keys %max;

my $nsolutions=0;


sub generate {
  my ($idx,$word)=@_;
  #pp \@_;

  return if $idx >= @keysort;

  my $newword;
  
  my $key= $keysort[$idx];
  for my $n (0..$max{$key}){
#    print "\t"x$idx ,"$n x $key\n";
    my $newword = $word.($key x $n);
 
    if ($maxword== length $newword) {
#      print "-> $newword\n";
      $nsolutions++;
      last;
    }

    generate($idx+1, $newword  )
  }
}

generate(0,"");

print $nsolutions;
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